Fractional Navier Stokes Equations Research Articles

An application of the iterative method to study multi-dimensional fractional order Navier-Stokes equations

In this article, a hybrid method called iteration Shehu transform method has been implemented to solve fractional-order Navier–Stokes equation. Atangana-Balenu operator describes fractional-order derivatives. The analytical solutions of three distinct examples of the time- fractional Navier-Stokes equations are determined by using Iterative shehu transform method. Further, we present the effectiveness and accuracy of the proposed method by comparison of analytical solutions to the exact solutions and the results are represented graphically and numerically.

On the Fuzzy Solution of Linear-Nonlinear Partial Differential Equations

In this article, we present the fuzzy Adomian decomposition method (ADM) and fuzzy modified Laplace decomposition method (MLDM) to obtain the solutions of fuzzy fractional Navier–Stokes equations in a tube under fuzzy fractional derivatives. We have looked at the turbulent flow of a viscous fluid in a tube, where the velocity field is a function of only one spatial coordinate, in addition to time being one of the dependent variables. Furthermore, we investigate the fuzzy Elzaki transform, and the fuzzy Elzaki decomposition method (EDM) applied to solving fuzzy linear-nonlinear Schrodinger differential equations. The proposed method worked perfectly without any need for linearization or discretization. Finally, we compared the fuzzy reduced differential transform method (RDTM) and fuzzy homotopy perturbation method (HPM) to solving fuzzy heat-like and wave-like equations with variable coefficients. The RDTM and HPM solutions are simpler than other already existing methods. Several examples are provided to illustrate the methods that have been offered. The results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. These studies are important in the context of the development of the theory of fuzzy partial differential equations.

Local L2 theory of the fractional Navier-Stokes equations and the self-similar solution

Local L2 theory of the fractional Navier-Stokes equations and the self-similar solution

Average Process of Fractional Navier–Stokes Equations with Singularly Oscillating Force

The averaging process between two-dimensional fractional Navier–Stokes equations driven by a singularly oscillating external force and the averaged equations corresponding to the limiting case are investigated. The uniform boundedness of the global attractors for a fractional Navier–Stokes equation with a singularly external force is established. Furthermore, these global attractors converge uniformly to the attractor of the averaged equations under suitable assumptions on the singularly external force, and the explicit convergence rate of the global attractors is guaranteed.

Mild Solutions of Time Fractional Navier-Stokes Equations Driven by Finite Delayed External Forces

Mild Solutions of Time Fractional Navier-Stokes Equations Driven by Finite Delayed External Forces

Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space dot{mathcal {B}}^{4-2alpha -frac{3}{p}}_{p,r}. Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.

Information technology of preschool education reform of fine arts based on fractional differential equation

Abstract Objective To study the information reform of preschool education of fine arts under fractional differential equation. Methods: In this paper, the fractional Navier-Stokes equations are compared with the traditional models through some numerical experiments, which provides a certain basis for future physical application. In order to improve students’ comprehensive ability and quality, we should strengthen education and teaching in all aspects, and art education is a more important aspect. The results show that the numerical solution of the integral equation with different values is close to a point at x = 1, that is, u (1) = 0.5. Conclusion In the current preschool education professional art education teaching process, the use of information technology means has become a necessary teaching requirement and need to improve the teaching effect, and the quality is of great significance and value, thus ensuring better able to carry out education of fine arts teaching, which enables students for better development, guarantees for a better future that is better for preschool education.

Fractional Navier–Stokes regularity criterion involving the positive part of the intermediate eigenvalue of the strain matrix

This paper proves the regularity criterion for the solutions of the 3D fractional Navier–Stokes equations in terms of the positive part of the intermediate eigenvalue of the strain matrix, which can be regarded as an improvement and extension of the results obtained by Miller [Arch. Ration. Mech. Anal. 235, 99–1139 (2020)] and by Wu [Evol. Equations Control Theory 10, 511–518 (2021)].

Invariant analysis, exact solutions and conservation laws of (2+1)-dimensional time fractional Navier–Stokes equations

In this paper, we investigate the exact solutions and conservation laws of (2 + 1)-dimensional time fractional Navier–Stokes equations (TFNSE). Specifically, Lie symmetries and corresponding one-dimensional optimal system for TFNSE in Riemann–Liouville sense are obtained. Then, based on the admitted symmetries and optimal system, we reduce these equations to one-dimensional equations or (1 + 1)-dimensional fractional partial differential equations (PDEs) with the help of Erdélyi–Kober fractional differential operator and compound variable transformation. In addition, we solve the reduced PDEs applying power series expansion method and invariant subspace method, respectively. Furthermore, the conservation laws of TFNSE are derived using new Noether theorem.

Numerical study for the unsteady space fractional magnetohydrodynamic free convective flow and heat transfer with Hall effects

In this paper, a numerical research for the problem of unsteady space fractional magnetohydrodynamic (MHD) free convective flow and heat transfer with Hall effects is investigated. We firstly establish a space fractional MHD flow and heat transfer model, which is coupled by the incompressible space fractional Navier–Stokes equations and the heat conduction equation. Then we develop a finite difference spectral decomposition method based on the pressure correction algorithm to solve this nonlinear coupled model. The validity of the proposed numerical scheme is verified and some flow field, temperature field are obtained. We also detailedly analyze the effects of relevant parameters on the space fractional MHD flow and heat transfer.

Axisymmetric Slow Motion of a Porous Spherical Particle in a Viscous Fluid Using Time Fractional Navier–Stokes Equation

In this paper, we present the unsteady translational motion of a porous spherical particle in an incompressible viscous fluid. In this case, the modified Navier–Stokes equation with fractional order time derivative is used for conservation of momentum external to the particle whereas modified Brinkman equation with fractional order time derivative is used internal to the particle to govern the fluid flow. Stress jump condition for the tangential stress along with continuity of normal stress and continuity of velocity vectors is used at the porous–liquid interface. The integral Laplace transform technique is employed to solve the governing equations in fluid and porous regions. Numerical inversion code in MATLAB is used to obtain the solution of the problem in the physical domain. Drag force experienced by the particle is obtained. The numerical results have been discussed with the aid of graphs for some specific flows, namely damping oscillation, sine oscillation and sudden motion. Our result shows a significant contribution of the jump coefficient and the fractional order parameter to the drag force.

Coriolis effect on temporal decay rates of global solutions to the fractional Navier–Stokes equations

The incompressible fractional Navier–Stokes equations in the rotational framework is considered. We establish the global existence result and temporal decay estimate for a unique smooth solution when the speed of rotation is sufficiently rapid. It is found that the strong rotational effect enhances the temporal decay rate of a certain norm of the velocity.

Partial Regularity for the Steady Hyperdissipative Fractional Navier–Stokes Equations

We extend the Caffarelli-Kohn-Nirenberg type partial regularity theory for the steady $5$-dimensional fractional Navier-Stokes equations with external force to the hyperdissipative setting. In our argument we use the methods of Colombo-De Lellis-Massaccesi to apply a blowup procedure adapted from work of Ladyzhenskaya-Seregin.

Dissipativity of Fractional Navier–Stokes Equations with Variable Delay

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problem.

Lagrange nodal discontinuous Galerkin method for fractional Navier-Stokes equations

This paper provides a Lagrange nodal discontinuous Galerkin method for solving the time-dependent incompressible space fractional Navier-Stokes equations numerically. The existence and uniqueness of weak solutions are obtained. By combining the Lagrange method in temporal discretization and the hybridized discontinuous Galerkin method in spatial direction, the fully discrete scheme is presented and the stability is proved rigorously. Furthermore, the error estimates for the L2-norm are derived in both the velocity and the pressure. Finally, some numerical experiments are given to illustrate the performance of the proposed method and validate the theoretical result.

Uniform analytic solutions for fractional Navier–Stokes equations

In this paper, we present the existence of the uniform analytic solution of the Cauchy problem for fractional incompressible Navier–Stokes Equations in critical Fourier–Herz spaces. The main strategies are to prove that the existence of the uniform analytic solution is equivalent to the boundedness of convolution inequality on Herz space.

Balance of the Vorticity Direction and the Vorticity Magnitude in 3D Fractional Navier–Stokes Equations

Fractional Navier–Stokes equations—featuring a fractional Laplacian-provide a ‘bridge’ between the Euler equations (zero diffusion) and the Navier–Stokes equations (full diffusion). The problem of whether an initially smooth flow can spontaneously develop a singularity is a fundamental problem in mathematical physics, open for the full range of models—from Euler to Navier–Stokes. The purpose of this work is to present a hybrid, geometric-analytic regularity criterion for solutions to the 3D fractional Navier–Stokes equations expressed as a balance—in the average sense—between the vorticity direction and the vorticity magnitude, key geometric and analytic descriptors of the flow, respectively.

Global existence of a uniformly local energy solution for the incompressible fractional Navier–Stokes equations

We introduce the concept of local Leray solutions starting from locally square-integrable initial data to the fractional Navier–Stokes equations with $s\in [3/4,1)$. Furthermore, we prove their local-in-time existence when $s\in (3/4, 1)$. In particular,

Generalized iterative method for the solution of linear and nonlinear fractional differential equations with composite fractional derivative operator

In present paper, we introduced generalized iterative method to solve linear and nonlinear fractional differential equations with composite fractional derivative operator. Linear/nonlinear fractional diffusion-wave equations, time-fractional diffusion equation, time fractional Navier-Stokes equation have been solved by using generalized iterative method. The graphical representations of the approximate analytical solutions of the fractional differential equations were provided.

Uniform well-posedness and stability for fractional Navier-Stokes equations with Coriolis force in critical Fourier-Besov-Morrey spaces

Uniform well-posedness and stability for fractional Navier-Stokes equations with Coriolis force in critical Fourier-Besov-Morrey spaces